The method used is based on the Augmented Lagrangian method commonly used to solve the Stokes equation for yield stress fluids
[1]}, [2]}. It is based on a variational method of the problem.
Indeed, if we rewrite the local equation (REF ) and introduce the function \(f(q)\)  as :
\(\delta p(q) = -f(q)=\left\lbrace \begin{array}{ll}- \frac{1}{\alpha } q & : |q|< q_c\; \\- \frac{1}{\beta } \left[ q - \frac{q}{|q|} ( 1 - \frac{\beta }{\alpha } ) \right] & : |q|>q_c\;,\\\end{array}\right.\) 
